Stuck on algebra's distributive properties [and] the following problem:
2a(b + 4) + 7(b + 4) …
Hello Rome86. You were asked to factor the given expression, right? (In the future, please include exercise instructions.)
Factoring and distributing are different -- but related -- processes. In fact, each "undoes" the effect of the other. Here's a simple example.
A(B + C)
That expression is factored because it shows two factors being multiplied. One factor is the symbolic number A, and the other factor is the symbolic number B+C. The distributive property tells us how to multiply them (that is, it tells us how to express the product of those two symbolic numbers): Each quantity inside the grouping symbols gets multiplied by the quantity outside.
A·B + A·C
That's the product of A times B+C. We say that the multiplication by A has been
distributed across the sum of B+C.
If we need to "undo" a distribution, then we factor. By inspection, we see that A is a common factor.
A·B +
A·C
To factor the expression effectively means to "pull out" the common factor and write it in front of what's left. We enclose what's left within grouping symbols, to show multiplication by A in factored form.
A(B+C)
Hopefully, that basic example helps you see why distributing and factoring are inverse processes (each undoes the other).
In your exercise, you've been given an expression that contains a common factor.
2a(
b + 4) + 7(
b + 4)
The common factor is the symbolic number b+4. We pull it out of the expression and write it in front of what's left. What's left is the symbolic number 2a+7.
(
b + 4)(2a + 7)
The book's answer shows the two factors written in reversed order. That's okay because the Commutative Property tells us that we may add or multiply quantities in whatever order we like (i.e., 2+3 is the same number as 3+2, and 2×3 is the same number as 3×2).
So, the answer may be written either way; both results express the same value.
(2a + 7)(b + 4)
(b + 4)(2a + 7)
?
. But i can't seem to figure out how to get there, which steps are taken and maybe more importantly why. If someone would be willing to explain the proces, that would be a big help.
